Supplementary Information for
Quantitative analysis of spin wave dynamics in ferrimagnets across compensation points
Eloi Haltz, Jo˜
ao Sampaio, Sachin Krishnia, L´
eo Berges, Rapha¨
el Weil, Alexandra Mougin, and Andr´
e Thiaville
I. REORIENTATION RANGE IN FERRIMAGNETS
In ferrimagnets with out-of-plane anisotropy, the range of5
in-plane field Hfor which the magnetization is aligned with
the field varies with temperature T. The Fig. S1 shows the
reorientation region for different values of the inter-lattice ex-
change (and for the same other parameters as in the main
text). In the limit of small Hin comparison with the inter-10
lattice exchange, the sublattice moments remain antiparallel
and the magnetization is out-of-plane for H<HK(T), where
HK(T)can be estimated with the formula for ferromagnets
(HK(T) = 2K
µ0MS(T)−MS(T)). Near TMC and for larger H,
the field induces a tilt between the two sublattices, creating15
a transverse net magnetic moment that aligns with H. This
pushes the two moments to be perpendicular to the field, in
a similar way to the spin-flop transition in antiferromagnets.
The border of the gray region can be obtained analytically, al-
though its formula is cumbersome. In our system, these two20
behaviors coexist, as is shown by the gray region in Fig. 3(e).
II. BRILLOUIN LIGHT SCATTERING
To measure the SW dynamics, we performed BLS spec-
trometry (see Fig. 1(b)), whereby a laser beam is backscat-
tered by the sample and its spectrum is analyzed. The sam-25
ple, installed on a temperature-controlled holder, is illumi-
nated by a laser beam linearly polarized in ˆ
y(s-polarized)
and the backscattered light polarized in the (ˆ
x,ˆ
z)plane (p-
polarized) is analyzed with a tandem Fabry-Perot spectrom-
eter (The Table Stable Ltd, model TFP-2 HC). Through the30
Stokes (S) and the anti-Stokes (AS) processes, the incident
photons can generate and absorb SW modes with a wave vec-
tor kSk −ˆ
xor kAS kˆ
x, respectively. The magnitude of kS/AS
is fixed by the angle of incidence θB=30° and the light wave-
length λ=532 nm such as −kS=kAS = (4πsin(θB)/λ)ˆ
x=35
(11.8µm−1)ˆ
x. The frequency of the backscattered light is
shifted by the SW frequency, producing peaks at fAS >0
and fS<0 in the BLS spectrum. A magnetic field H
along ˆ
y(Fig. 1(b)) is applied to counteract the perpendicular
anisotropy and measure the SW in the Damon-Eshbach geom-40
etry (kkˆ
xand mkˆ
y) . This geometry gives a larger signal,
allows easily applying a magnetic field, and is convenient to
measure the SW chirality [17,18].
The points in the center of the spectra ( f≈0) correspond
to the reference unscattered light and were not considered in45
the analysis. All shown spectra were normalized to their max-
imum point.
In some spectra at the lowest temperatures, the HF peaks
do not appear due to the restricted measurement range of
±30 GHz instead of ±100 GHz. This was chosen to reduce50
TMC
H
mRE mTM
spin-
flop
H < HK
H > HK
mz,TM
1
0
J = 80 MJ/m³
µ₀H [T]µ₀H [T]µ₀H [T]
T [K]
40 MJ/m³
20 MJ/m³
Figure S1. Diagram of the reorientation range versus in-plane field
Hand temperature Tin a RE-TM ferrimagnetic alloy for inter-lattice
exchange Jvalues above, at, and below the fitted value. The gray
background corresponds to the out-of-plane component of the TM
magnetization. The dotted line shows HK(T)using the effective
ferromagnet formula (equal in the three plots). The dashed line
shows the spin-flop boundaries HJ(T)calculated by neglecting the
anisotropy (different in the three plots). Vertical lines correspond to
TMC and TAC.
the instrumental broadening when measuring the sharper LF
peaks (instrumental full-width/half-maximum of ∼1.2 GHz
instead of ∼4 GHz).
III. ADDITIONAL DATA
Fig. S2 shows AHE hysteresis cycles measured in a quasi-55
in-plane geometry at different temperatures, used to determine
2
Figure S2. Anomalous Hall effect loops under field in quasi-in-plane
geometry (83° to the normal of the film) for different temperatures.
HK. The AHE signals were measured in a separate sample
of the film, which was lithographically patterned into Hall
crosses. After the patterning process, the TMC of the AHE
sample decreased by 30 K. This decrease was corrected in the
temperature values of the HKdata.5
Figs. S3 and S4 show additional measurements of the mode
frequencies versus temperature or field, similar to main text
but at different constant field or temperature, respectively.
3
Figure S3. Top panel: BLS peak centers and widths versus T for different applied fields H. Similarly to Fig. 2(a), peak centers f0(points) and
widths δ(vertical bars) are extracted from fits. Lines and shaded regions are the frequency and peak width calculated with the Smit-Beljers
method. Bottom panel: Frequency difference ∆f=|fAS|−|fS|for the LF and HF peaks versus T similarly to Fig. 2(d). The lines are the
prediction of the Smit-Beljers method. The data for H=0.56 T are the same shown Fig. 2.
Figure S4. Top panel: BLS peak centers and widths versus H for different temperatures. Similarly to Fig 3(a), peak centers f0(points) and
widths δ(vertical bars) are extracted from fits. Lines and shaded regions are the frequency and peak width calculated with the Smit-Beljers
method. Bottom panel: Frequency difference ∆f=|fAS|−|fS|for the LF and HF peaks versus H similarly to Fig 3(b). The lines are the
prediction of the Smit-Beljers method. The data for T=367 K are the same shown in Fig 3(a,b).
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